The results of such a modelizing method can be used to prepare safety analysis reports before building and starting a reactor.
These results can also be useful for existing nuclear reactors and especially for managing the nuclear fuel loaded therein. In particular, these results can be used to assess how the core design should evolve in time and decide of the positions of the fuel assemblies in the core, especially the positions of the fresh assemblies to be introduced in the core.
Such modelizing methods are implemented by computers. To this end, the core is partitioned in cubes, each cube constituting a node of a grid for implementing a digital computation.
Usually the steady-state diffusion equation to be solved during such a digital computation amounts to:
                                                                        Σ                Rg                m                            ⁢                              ϕ                g                m                                                    ︸              removal                                =                                                    λχ                g                            ⁢                                                ∑                                                            g                      ′                                        =                    1                                    G                                ⁢                                                      ν                    ⁢                                                                                  ⁢                                          Σ                                              fg                        ′                                            m                                        ⁢                                          ϕ                                              g                        ′                                            m                                                                            ︸                    production                                                                        +                                          ∑                                                      g                    ′                                    ≠                  g                                G                            ⁢                                                                    Σ                                          gg                      ′                                        m                                    ⁢                                      ϕ                                          g                      ′                                        m                                                                    ︸                  inscatter                                                      +                                          ∑                                                      u                    =                    x                                    ,                  y                  ,                  z                                            ⁢                                                1                                      a                    u                    m                                                  ⁢                                                      θ                    gu                    m                                    ⁡                                      (                                                                                            j                          gul                                                      +                            m                                                                          +                                                  j                          gur                                                      -                            m                                                                                                                      ︸                        incurrent                                                              )                                                                                      ,                            (        1        )                                                          ⁢                  with          ⁢                      {                                                                                Σ                    Rg                    m                                                                    =                                                                                            Σ                      ag                      m                                        +                                                                  ∑                                                                              g                            ′                                                    ≠                          g                                                G                                            ⁢                                              Σ                                                                              g                            ′                                                    ⁢                          g                                                m                                                              +                                                                  ∑                                                                              u                            =                            x                                                    ,                          y                          ,                          z                                                                    ⁢                                                                        2                          ⁢                                                                                                          ⁢                                                      c                                                          1                              ⁢                              gu                                                        m                                                                                                    a                          u                          m                                                                                                                                                                                      θ                    gu                    m                                                                    =                                                                                            1                      -                                              c                                                  2                          ⁢                                                                                                          ⁢                          gu                                                m                                            -                                              c                                                  3                          ⁢                                                                                                          ⁢                          gu                                                m                                                              ,                                                                                                          (        2        )            
where λ is a first neutron eigenvalue, m is a cube index, also called nodal index, G is the number of neutron energy groups and g, g′ are neutron energy group indexes, u is a Cartesian axis index of the cube, Σagm represents macroscopic absorption cross-section for the cube m and the energy group g, Σfg′m represents macroscopic fission cross-section for the cube m and the energy group g′, Σgg′m represents macroscopic slowing down cross-section for the cube m and the energy groups g, Φgm represent neutron fluxes, such that the ΣRgm, . . . Φgm represent the reaction rates for the corresponding reactions (absorption, fission), v is the number of neutrons produced per fission, χg is the fraction of neutrons emerging from fission with neutron energy g, aum is the width of cube m along Cartesian axis u, and
with the relationship between the neutron outcurrents jgul−m and jgur+m, neutron fluxes Φgm and neutron incurrents jgul+m and jgur−m defined by:
                    {                                                                              j                  gul                                      -                    m                                                  =                                                                            c                                              1                        ⁢                                                                                                  ⁢                        gu                                            m                                        ⁢                                          ϕ                      g                      m                                                        +                                                            c                                              2                        ⁢                                                                                                  ⁢                        gu                                            m                                        ⁢                                          j                      gul                                              +                        m                                                                              +                                                            c                                              3                        ⁢                                                                                                  ⁢                        gu                                            m                                        ⁢                                          j                      gur                                              -                        m                                                                                                                                                                                      j                  gur                                      +                    m                                                  =                                                                            c                                              1                        ⁢                                                                                                  ⁢                        gu                                            m                                        ⁢                                          ϕ                      g                      m                                                        +                                                            c                                              3                        ⁢                                                                                                  ⁢                        gu                                            m                                        ⁢                                          j                      gul                                              +                        m                                                                              +                                                            c                                              2                        ⁢                                                                                                  ⁢                        gu                                            m                                        ⁢                                          j                      gur                                              -                        m                                                                                                                                                    (        3        )            
The coefficients cigum, with i=1, 2, 3, are characteristic of the cube in and depend on nodal dimensions and macroscopic cross-sections Σm.
FIG. 1 is a schematic representation in two dimensions of a cube m showing the neutron incurrents jgul+m and jgur−m for u=x, y and z; the neutron outcurrents jgul−1 and jgur+m for u=x, y and z; and the neutron fluxes Σgm. Indexes 1, respectively r, refers to each left interface surface, respectively each right interface surface, of the cube m for the respective Cartesian axis x, y. Indexes +, respectively −, represents the orientation from left to right, respectively from right to left, for the respective Cartesian axis x, y.
The steady-state diffusion equation (1) is also named NEM equation, for Nodal Expansion Method equation.
In the state of the art methods, most of the computational efforts are concentrated in the part dedicated to the iterative solving of a large eigensystem corresponding to the steady-state diffusion equation (1).
In order to lower these computational efforts and therefore accelerate the solving of the eigensystem, Coarse Mesh Rebalancing (CMR) procedures have been used. In these procedures, neutron fluxes and currents for a given iteration are multiplied with a corrective factor before pursuing subsequent computationally expensive iterations. The multiplicative correction serves to suppress the presence of a non fundamental wavelength part of eigenspectrum with the first neutron eigenvalue λ close to an exact value λexact.
However, the acceleration effect realized in this way depends on the numerical proximity of the highest coarse mesh level in a multi-level hierarchy to the full-core diffusion level. Such CMR procedures may therefore lead to very slow convergence or even convergence stagnation, thus increasing the computational effort.